Relative categoricity in abelian groups. II (Q1012332)

[For Part I by \textit{W. Hodges} see: S. B. Cooper et al. (eds.), Models and computability. Invited papers from the Logic colloquium '97, European meeting of the Association for Symbolic Logic, Leeds, UK, July 6--13, 1997. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 259, 157--168 (1999; Zbl 0958.03024).] Let \(L\) be a first-order language, and \(L(P)\) be \(L\) expanded with an additional unary relation symbol \(P\). Consider \(L(P)\)-structures \(A\) such that the set of elements \(A^p\) of \(A\) satisfying \(P(x)\) is an \(L\)-substructure of the (reduct to \(L\)) of \(A\). Such an \(A\) is said to be \((\kappa,\lambda)\)-categorical if for all models \(B\), \(C\) of its theory (the set of \(L(P)\)-sentences true in \(A\)), if furthermore \(|B^P|=|C^P|=\kappa\), \(|B|=|C|=\lambda\) and \(B^P=C^P\), then there is an isomorphism from \(B\) to \(C\) which extends \(B^P=C^P\). This paper considers the case of abelian groups, i.e., \(L\) is the language with \(+,-\) and \(0\), every considered \(L(P)\)-structure \(A\) is an abelian group and \(A^p\) is a subgroup of \(A\). Such an \(A\) is said to be an abelian group pair. The paper classifies the possible pairs \((\kappa,\lambda)\) for which an abelian group pair can be \((\kappa,\lambda)\)-categorical (the relative categoricity spectra). The authors show that for \(A^P\) infinite, the possible spectra are either all \((\kappa,\kappa)\) for \(\kappa\) an infinite cardinal, all \((\kappa,\lambda)\) where either \(\omega\leq \kappa < \lambda\) or \(\omega=\kappa=\lambda\), the single pair \((\omega,\omega)\) or, finally, all pairs such that \(\omega\leq \kappa < \lambda\). They furthermore give an algebraic description in each case. Finally, they also give a characterisation of relative categoricity spectra when \(A^p\) is finite and also when both \(A\) and \(A^p\) are finite. The proofs use methods from abelian group theory, model theory of abelian group and in some cases general model theory.